Definition:Conditional Expectation/General Case/Event

Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be an integrable random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $A \in \Sigma$.

Then we define the conditional expectation of $X$ given $A$:

$\expect {X \mid A} = \expect {X \mid \map \sigma A}$

where:

$\map \sigma A$ denotes the $\sigma$-algebra generated by $A$

$\expect {X \mid \map \sigma A}$ denotes the conditional expectation of $X$ given $\map \sigma A$

$=$ is understood to mean almost-sure equality.